Computing Rational Expectations An AR(2) Example

Let us consider the following AR(2) process xt = ϕ1 xt−1 + ϕ2 xt−2 + εt

(1)

where εt ; N (0, σ 2 ) is the innovation of the process and where ϕ1 and ϕ2 are such that the roots lies outside the unit circle. The aim of this note is to understand how agents compute the rational expectation of xt , E[xt |Ω], for various information sets Ω. The agent is assumed to know the model (Equation 1), the coefficients (ϕ1 , ϕ2 ), and the distribution of the shocks. Let us first specify Ω = {xk ; k = 0, . . . , t − 1}. Our aim is to compute the rational expectation E[xt |Ω], as formed by an agent. First of all, since the agent knows the model, she can use the fact that E[xt |Ω] = E[ϕ1 xt−1 + ϕ2 xt−2 + εt |Ω] By linearity of the conditional expectation operator, we then have E[xt |Ω] = E[ϕ1 xt−1 |Ω] + E[ϕ2 xt−2 |Ω] + E[εt |Ω] Since the agents know the parameters ϕ1 and ϕ2 , these two parameters are not a random variable from the point of view of the agents and are then treated as constants which can be taken out of the expectation operator to lead to E[xt |Ω] = ϕ1 E[xt−1 |Ω] + ϕ2 E[xt−2 |Ω] + E[εt |Ω] Note that, by construction of the information set Ω, we have xt−1 ∈ Ω and xt−2 ∈ Ω. This implies that the agent observes both xt−1 and xt−2 which therefore do not need to be forecasted. In other words, we have E[xt−1 |Ω] = xt−1 and E[xt−2 |Ω] = xt−2 , such that E[xt |Ω] = ϕ1 xt−1 + ϕ2 xt−2 + E[εt |Ω] Since, εt is an innovation, it is orthogonal to any past realization of the process, εt ⊥Ω. The best prediction the agent can make is therefore its unconditional expectation such that E[εt |Ω] = 0. Hence, the rational expectation of xt conditional on the information set Ω is given by E[xt |Ω] = ϕ1 xt−1 + ϕ2 xt−2 1

Let us now assume that the information set, Ω, consists of all the past history of xt until period t − 2 rather than t − 1. In other words, we assume Ω = {xk ; k = 0, . . . , t − 2}. Using the fact that the agent knows the form of the model as well as the coefficients, we again have that E[xt |Ω] = ϕ1 E[xt−1 |Ω] + ϕ2 E[xt−2 |Ω] + E[εt |Ω] Like in the previous case, εt being an innovation, it is orthogonal to any past realization of the process and the best prediction that the agent can make is the unconditional mean, 0, implying E[xt |Ω] = ϕ1 E[xt−1 |Ω] + ϕ2 E[xt−2 |Ω]

(2)

xt−2 belongs the information set Ω, and is therefore observed by the agent, implying that E[xt−2 |Ω] = xt−2 . This is not the case for xt−1 that does not belong Ω, hence the agent needs to formulate an expectation for xt−1 . Since the agent knows that xt is generated by the process defined in Equation (1), she knows it applies for any period t; in particular in t − 1 xt−1 = ϕ1 xt−2 + ϕ2 xt−3 + εt−1 The conditional expectation of xt−1 is given by E[xt−1 |Ω] = E[ϕ1 xt−2 + ϕ2 xt−3 + εt−1 |Ω] For the same reasons as above (linearity, known coefficients), we have E[xt−1 |Ω] = ϕ1 E[xt−2 |Ω] + ϕ2 E[xt−3 |Ω] + E[εt−1 |Ω] Since both xt−2 and xt−3 are in Ω, we have E[xt−2 |Ω] = xt−2 and E[xt−3 |Ω] = xt−3 . However, εt−1 being orthogonal to Ω, we have E[xt−2 |Ω] = 0. Therefore, E[xt−1 |Ω] = ϕ1 xt−2 + ϕ2 xt−3 Using this result in (2), the rational expectation of xt conditional on Ω is given by E[xt |Ω] = (ϕ21 + ϕ2 )xt−2 + ϕ1 ϕ2 xt−3

2